\newcommand \nnn {{\noindent}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Environment
\newenvironment{pf}{\par\noindent{\em Proof}.}{\hfill\framebox(6,6)\par\medskip}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  Theorems
%%%%%%%%%%%%%%%%%%%%%%%%%%  Facts
\newtheorem{theorem}{Theorem}[section]
\newtheorem*{thr}{Theorem}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem*{prop}{Proposition}%%%%%%%%%%%%%%%%%%%%%%
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{conclusion}[theorem]{Conclusion}%added
\newtheorem{summary}[theorem]{Summary}%added
\newtheorem{fact}[theorem]{Fact}
\newtheorem{assumption}[theorem]{Assumption}
\newtheorem{claim}[theorem]{Claim}%added
%%%%%%%%%%%%%%%%%%%%%%%%%%  Examples
\newtheorem{example}[theorem]{Example}
\newtheorem{counterexample}[theorem]{Counter Example}%added
\newtheorem{exercise}[theorem]{Exercise}%added
%%%%%%%%%%%%%%%%%%%%%%%%%%  Questions
\newtheorem{problem}[theorem]{Problem}%added
\newtheorem{question}[theorem]{Question}
\newtheorem{crucialquestion}[theorem]{Crucial Question}
%%%%%%%%%%%%%%%%%%%%%%%%%%  Notation
\newtheorem{definition}[theorem]{Definition}
\newtheorem*{terminology}{Terminology} %added
%%%%%%%%%%%%%%%%%%%%%%%%%%  Methods 
\newtheorem{method}[theorem]{Method}%added
%%%%%%%%%%%%%%%%%%%%%%%%%%  Acknowledgement
\newtheorem{acknowledgement}{Acknowledgement}%added
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  Calligraphic
\newcommand \bcA {{\mathscr A}}
\newcommand \bcB {{\mathscr B}}
\newcommand \bcC {{\mathscr C}}
\newcommand \bcD {{\mathscr D}}
\newcommand \bcE {{\mathscr E}} %Sections
\newcommand \bcF {{\mathscr F}} %Sheaf
\newcommand \bcG {{\mathscr G}}
\newcommand \bcH {{\mathscr H}}
\newcommand \bcI {{\mathscr I}}
\newcommand \bcJ {{\mathscr J}}
\newcommand \bcK {{\mathscr K}}
\newcommand \bcL {{\mathscr L}}
\newcommand \bcM {{\mathscr M}}
\newcommand \bcN {{\mathscr N}}
\newcommand \bcO {{\mathscr O}} %Regular functions
\newcommand \bcP {{\mathscr P}}
\newcommand \bcQ {{\mathscr Q}}
\newcommand \bcR {{\mathscr R}}
\newcommand \bcS {{\mathscr S}}
\newcommand \bcT {{\mathscr T}}
\newcommand \bcU {{\mathscr U}}
\newcommand \bcV {{\mathscr V}}
\newcommand \bcW {{\mathscr W}}
\newcommand \bcX {{\mathscr X}}
\newcommand \bcY {{\mathscr Y}}
\newcommand \bcZ {{\mathscr Z}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  Double Strike
\newcommand \ZZ {{\mathds Z}} % integers %Changed
\newcommand \FF {{\mathds F}} % Finite Field %Changed
\newcommand \NN {{\mathds N}} % natural numbers %Changed
\newcommand \QQ {{\mathds Q}} % rational numbers %Changed
\newcommand \RR {{\mathds R}} % real numbers %Changed
\newcommand \CC {{\mathds C}} % complex numbers %Changed
\newcommand \PR {{\mathds P}} % projective space %Changed
\newcommand \AF {{\mathds A}} % affine space %Changed
\newcommand \VV {{\mathds V}} % Variety %Added
\newcommand \II {{\mathds I}} % Ideal of a variety %Added
\newcommand \KK {{\mathds K}} % Field %Added
\newcommand \kk {{\mathds k}} % Field %Added
\newcommand \rf {{\mathbf k}} % Residue field %Added
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  Double Strike
\newcommand \HH {{\mathds H}} % added
\newcommand \uno {{\mathbbm 1}}
\newcommand \Le {{\mathbbm L}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  Roman
\newcommand \Alb {{\rm {Alb}}}	
\newcommand \alg {{\rm {alg}}}	
\newcommand \Alt {{\rm {Alt}}}	
\newcommand \Aut {{\rm Aut}}	
\newcommand \Bl {{\rm {Bl}}}	
\newcommand \cha {{\rm {char}}}	
\newcommand \codim {{\rm {codim}}}	
\newcommand \coim {{\rm {coim}}}	% added
\newcommand \Coim {{\rm Coim}}	% added
\newcommand \coker {{\rm {coker}}}	
\newcommand \Coker {{\rm {Coker}}}	% added
\newcommand \colim {{{\rm colim}\, }} % colimit	
\newcommand \Colim {{{\rm Colim}\, }} % colimit	
\newcommand \cone {{\rm {cone}}}	
\newcommand \cor {{\rm {cor}}}	
\newcommand \Corr {{Corr}}	
\newcommand \dom {{dom}}	
\newcommand \codom {{codom}}	% added
\newcommand \End {{\rm {End}}}	
%\newcommand \fix {{\rm {fix}}}	% added
\newcommand \Gal {{\rm {Gal}}}	
\newcommand \Gr {{\rm {Gr}}}	
\newcommand \Hom {{\rm Hom}}	
\newcommand \id {{\rm {id}}}	
\newcommand \im {{\rm im}}	
\newcommand \Imm {{\rm Im}}	% added
\newcommand \inv {{\rm {inv}}}	
%\newcommand \Jac {{\rm {Jac}}}	% added
\newcommand \Jac {{{J}}}	
%!%\newcommand \ker {{\rm {ker}}}	% added
\newcommand \Ker {{\rm {Ker}}}	% added
%\newcommand \mob {{\rm {mob}}}	% added
\newcommand \NS {{{NS}}}	
\newcommand \Ob {{\rm Ob}}	
%\newcommand \ord {{\rm {ord}}}	% added
\newcommand \PGL {{\rm {PGL}}}	
\newcommand \Pic {{\rm {Pic}}}	
\newcommand \Proj {{\rm {Proj}}}	% added
\newcommand \Prym {{\rm {Prym}}}	
\newcommand \ptr {{\pi _2^{\rm tr}}}	
\newcommand \rad {{\rm {rad}}}	% added
\newcommand \Rad {{\rm {Rad}}}	% added
\newcommand \res {{\rm {res}}}	
\newcommand \Sing {{\rm {Sing}}}	
\newcommand \spc {{\rm {sp}}}	
\newcommand \Spec {{\rm {Spec}}}	
\newcommand \spn {{\rm {span}}}	
%\newcommand \supp {{\rm {supp}}}	% added
\newcommand \Sym {{\rm {Sym}}}	
\newcommand \tr {{\rm {tr}}}	
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  Gothic
\newcommand \goa {{\mathfrak a}}% added
\newcommand \gob {{\mathfrak b}}% added
\newcommand \gom {{\mathfrak m}}% added
\newcommand \gon {{\mathfrak n}}% added
\newcommand \goM {{\mathfrak M}}% added
\newcommand \goN {{\mathfrak N}}% added
\newcommand \gop {{\mathfrak p}}% added
\newcommand \goq {{\mathfrak q}}% added
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  Categories
\newcommand \SH {{\mathbf{SH}}}%added
\newcommand \PreC {{\mathbf{PreShv}}^{\bcA}(\bcC)}%added
\newcommand \Pre {{\mathbf{PreShv}}^{\bcA}}%added
\newcommand \ShvC {{\mathbf{Shv}}^{\bcA}_{\tau}(\bcC)}%added
\newcommand \Shv {{\mathbf{Shv}}^{\bcA}}%added
\newcommand \Cov {{\mathbf{Cov}}_{\tau}}%added
\newcommand \Sets {{\mathscr S\!\! ets}}%added
\newcommand \SSets {{\mathscr S\!\! \mathscr S\!\! ets}}%added
\newcommand \FSets {{\mathscr F\!\!\! \mathscr S\!\! ets}}%added
\newcommand \Simp {{\mathbf{\Delta}}}%added
\newcommand \Top {{\mathscr T\!\! op}}%added
\newcommand \Mod {{\!-\!\mathbf{Mod}}}%added
\newcommand \Vct {{\mathbf{Vect}}}%added
\newcommand \FVct {{\mathbf{FVect}}}%added
\newcommand \Grp {{\mathbf{Grp}}}%added
\newcommand \Ab {{\mathbf{Ab}}}%added
\newcommand \Svs {{\mathbf{Sieves}}}%added
\newcommand \Sch {{\mathbf{Sch}}}%added
\newcommand \Sm {{\mathbf{Sm}}}%added
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  Arrows
\newcommand \lra {\longrightarrow}
\newcommand \hra {\hookrightarrow}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  Colours
\def\blue {\color{blue}}
\def\red {\color{red}}
\def\green {\color{green}}
\def\black {\color{black}} %added
\definecolor{dark-gray}{rgb}{0.5,0.5,0.5} %added
\def\gray {\color{dark-gray}} %added
\newcommand \tc \textcolor{}{} %added
\newcommand \tcr {\textcolor{red}}{} %added
\newcommand \tcb {\textcolor{blue}}{} %added
\newcommand \tcg {\textcolor{green}}{} %added
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  Combined symbols
\newcommand \CHM {{\mathscr C\! \mathscr M}}
\newcommand \DM {{\sf DM}}
\newcommand \FS {{FS}}
\newcommand \MM {{\mathscr M\! \mathscr M}}
\newcommand \HS {{\mathscr H\! \mathscr S}}
\newcommand \MHS {{\mathscr M\! \mathscr H\! \mathscr S}}
\newcommand \Vect {{\mathscr V\! ect}} %
\newcommand \Gm {{{\mathbb G}_{\rm m}}}
\newcommand \trdeg {{\rm {tr.deg}}}
\newcommand \znak {{\natural }}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  ???
\newcommand \goT {{\mathfrak T}}
\newcommand \goC {{\mathfrak C}}
\newcommand \goD {{\mathfrak D}}
\newcommand \goS {{\mathfrak S}}
\newcommand \goH {{\mathfrak H}}
\newcommand \goi {{\mathfrak i}}%added